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Quantum Algorithms for Element Distinctness
 SIAM Journal of Computing
, 2001
"... We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison c ..."
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Cited by 75 (9 self)
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We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with Θ(N log N) classical complexity. We also prove a lower bound of Ω ( √ N) comparisons for this problem and derive bounds for a number of related problems. 1
TimeSpace Tradeoffs for Branching Programs
, 1999
"... We obtain the first nontrivial timespace tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant & ..."
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Cited by 47 (4 self)
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We obtain the first nontrivial timespace tradeoff lower bound for functions f : {0, 1}^n &rarr; {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + &epsilon;)n, for some constant &epsilon; > 0. We also give the first separation result between the syntactic and semantic readk models [BRS93] for k > 1 by showing that polynomialsize semantic readtwice branching programs can compute functions that require exponential size on any syntactic readk branching program. We also show...
Determinism versus NonDeterminism for Linear Time RAMs with Memory Restrictions
 In Proc. of 31st STOC
, 1998
"... Our computational model is a random access machine with n read only input registers each containing c log n bits of information and a read and write memory. We measure the time by the number of accesses to the input registers. We show that for all k there is an epsilon > 0 so that if n is suffici ..."
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Cited by 41 (2 self)
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Our computational model is a random access machine with n read only input registers each containing c log n bits of information and a read and write memory. We measure the time by the number of accesses to the input registers. We show that for all k there is an epsilon > 0 so that if n is sufficiently large then the elements distinctness problem cannot be solved in time kn with epsilon n bits of read and write memory, that is, there is no machine with this values of the parameters which decides whether there are two different input registers whose contents are identical. We also show that there is a simple decision problem that can be solved in constant time (actually in two steps) using nondeterministic computation, while there is no deterministic linear time algorithm with epsilon n log n bits read and write memory which solves the problem. More precisely if we allow kn time for some fixed constant k, then there is an epsilon > 0 so that the problem cannot be solved with epsilon n log n bits of read and write memory if n is sufficiently large. The decision problem is the following: "Find two different input registers, so that the Hamming distance of their contents is at most c log n".
SuperLinear TimeSpace Tradeoff Lower Bounds for Randomized Computation
, 2000
"... We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, ..."
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Cited by 33 (2 self)
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We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, Ajt99b] in his timespace tradeoffs for deterministic RAM algorithms computing element distinctness and for Boolean branching programs computing a natural quadratic form. Ajtai's bounds were of the following form...
Quantum timespace tradeoffs for sorting
 Proceedings of 35th ACM STOC
, 2003
"... We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We o ..."
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Cited by 8 (2 self)
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We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds S, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T = O(n
Nondeterministic LinearTime Tasks May Require Substantially Nonlinear Deterministic . . .
, 1989
"... ..."
Separable Attributes: a Technique for Solving the Sub Matrices Character Count Problem
"... The subsequence character count problem has as its input an array S = s 1 ; :::; s n of symbols over alphabet and a natural number m. Its output is: for every i; i = 1; :::; n m + 1; the number of dierent alphabet symbols occurring in the subsequence s i ; s i+1 ; :::; s i+m 1 . The subsequence ..."
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Cited by 5 (2 self)
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The subsequence character count problem has as its input an array S = s 1 ; :::; s n of symbols over alphabet and a natural number m. Its output is: for every i; i = 1; :::; n m + 1; the number of dierent alphabet symbols occurring in the subsequence s i ; s i+1 ; :::; s i+m 1 . The subsequence character count problem is a natural problem that has many uses. It can be solved in linear time for xed nite alphabets and in time O(n log m) for innite alphabets. The character count problem can be generalized to two dimensions and becomes the submatrix character count problem. Its input is an n n matrix T over alphabet and a natural number m. Its output is: for every i; j; i; j = 1; :::; n m+ 1; the number of dierent alphabet symbols occurring in the submatrix T [i + k; j + `]; k = 0; :::; m 1; ` = 0; :::; m 1. The straightforward one dimensional solution slides a window along the text adding an element and deleting an element at every step. The problem with two dimensions is that at every move of the window there are m elements added and m deleted. In this paper we present an alternate one dimensional solution that generalizes to two dimensions. We achieve a O(n 2 ) time solution to the submatrix character count problem over nite xed alphabet and a O(n 2 log m) solution over an innite alphabet. 1